Introduction to Gaussian Process

Author: Matheus de Oliveira Schossler

Date: November 24, 2024

Gaussian process regression is a well-known method in Bayesian statistics. A Gaussian process (GP) is used to define a probability distribution over functions. It is inherently a Bayesian approach because it defines a prior distribution over functions. The prior is specified using the mean function and a covariance function, also called a kernel, which encodes our beliefs about the underlying function before observing any data. With this concept, we can build the Gaussian process regression algorithm (GPR), which is a data-based prediction method with good performance in dealing with high-dimensional small samples and nonlinear problems. The predictive mean and variance are derived by conditioning the prior GP on the observed data.

In this report, we will define and demonstrate an algorithm that facilitates the use of Gaussian processes. We will build our intuition from simpler regression algorithms that are instances of Gaussian process regressors, and then provide a few simple examples using the GPR language. Additionally, we will showcase an application of Gaussian process regression on a real-world dataset.

References

1. Rasmussen, C. E. & Williams, C. K. I. (2006). Gaussian Processes for Machine Learning. Massachusetts Institute of Technology. ISBN: 026218253X. URL.
2. McClarren, R. G. (2018). Uncertainty Quantification and Predictive Computational Science. Springer. DOI.
3. Jie Wang, An Intuitive Tutorial to Gaussian Processes Regression
4. CBE 67701: Uncertainty Quantification.
5. Gaussian Process Regression Tutorial.
6. Ng, A., Updated by Ma, T. (2022). CS229 Lecture Notes. Stanford University.
7. PyMC Gaussian Processes Tutorial.
8. PyMC Examples: Gaussian Processes - Latent.
9. PyMC Examples: Gaussian Processes - Marginal
10. Bayesian Regression Tutorial by Juanito Rdz
11. gaussian_processes/HSGP-Basic

Other references are indicated in the text.

from IPython.display import display, HTML, Image

display(HTML("<style>.container { width:80% !important; }</style>"))

Gaussian Process

A Gaussian Process (GP) is a framework used to define probability distributions over functions, offering a Bayesian probabilistic perspective. A Gaussian process extends the concept of a Gaussian probability distribution. While a probability distribution characterizes random variables that are either scalars or vectors (as in multivariate distributions), a stochastic process defines the behavior of functions. This framework allows us to use GPs as priors in function space, which is particularly advantageous in Bayesian statistics. The Gaussian Process Regression (GPR) algorithm, derived from this concept, excels in predicting outcomes from high-dimensional, small-sample, and nonlinear datasets [Ref. 1].

A GP is essentially a collection of random variables, any finite subset of which will have a jointly Gaussian distribution [Ref. 1]. This property ensures that the joint distributions are consistent across different subsets. For instance, if the joint distribution of $f(x_1), f(x_2), \ldots, f(x_n)$ is known, the distribution of a smaller subset $f(x_1), f(x_2)$ should correspond to what would be derived by marginalizing out $f(x_3), \ldots, f(x_n)$ from the full set.

A Gaussian process is fully characterized by its mean function $m(x)$ and covariance function $\Sigma_{ij} = k(x, x')$:

\begin{equation} f \sim \mathcal{GP}(m, k), \tag{1.1} \end{equation}

This notation indicates that the function $f$ is distributed as a Gaussian Process with mean function $m$ and covariance function $k$. The vector notation for a set of indices $\{x_i \in {X}\}_{i=1}^m$ representing the variables is denoted by $\boldsymbol{f}$:

\begin{equation} \left[ \begin{array}{c} f(x_1) \\ \vdots \\ f(x_m) \end{array} \right] \sim \mathcal{N}\left( \left[ \begin{array}{c} m(x_1) \\ \vdots \\ m(x_m) \end{array} \right], \left[ \begin{array}{ccc} k(x_1, x_1) & \cdots & k(x_1, x_m) \\ \vdots & \ddots & \vdots \\ k(x_m, x_1) & \cdots & k(x_m, x_m) \end{array} \right] \right), \tag{1.2} \end{equation}

where $\mathcal{N}(m, \Sigma)$ denotes a multivariate Gaussian distribution with mean vector $m$ and covariance matrix $\Sigma$, as previously defined.

Examples

Numerous models frequently used in both machine learning and statistics are actually specific instances of, or variations on, Gaussian processes. In this discussion, we will provide a few examples to illustrate the concept of Gaussian processes.

Bayesian Linear Regression

Consider a training set $S = \{(x^{(i)}, y^{(i)})\}_{i=1}^m$ of i.i.d. examples drawn from an unknown distribution. Linear regression models the relationship between $x^{(i)}$ and $y^{(i)}$ as:

$$ y^{(i)} = \theta^T x^{(i)} + \varepsilon, \tag{2.1} $$

where $\varepsilon$ represents i.i.d. noise following a normal distribution $\mathcal{N}(0, \sigma^2)$. This relationship leads to the likelihood function:

\begin{equation} p(y^{(i)} | x^{(i)}, \theta) = \frac{1}{\sqrt{2\pi} \sigma} \exp\left(-\frac{(y^{(i)} - \theta^T x^{(i)})^2}{2\sigma^2}\right). \tag{2.2} \end{equation}

In Bayesian linear regression, we introduce a prior distribution over the parameters $\theta$:

\begin{equation} \theta \sim \mathcal{N}(0, \tau^2 I), \tag{2.3} \end{equation}

Utilizing Bayes' rule, the posterior distribution of $\theta$ given the training set $S$ is defined as:

\begin{equation} p(\theta | S) = \frac{p(\theta) p(S | \theta)}{\int_{\theta'} p(\theta') p(S | \theta') \, d\theta'}. \tag{2.4} \end{equation}

Here, $p(S | \theta) = \prod_i p(y^{(i)} | x^{(i)}, \theta)$, where each $p(y^{(i)} | x^{(i)}, \theta)$ is the normal distribution defined in equation (1.2). The prediction for a new test point $x_*$ using this model is characterized by the posterior predictive distribution:

\begin{equation} p(y_* | x_*, S) = \int_\theta p(y_* | x_*, \theta) p(\theta | S) \, d\theta. \tag{2.5} \end{equation}

For the linear regression model described above, both $p(\theta | S)$ and $p(y_* | x_*, S)$ can be expressed in terms of a closed-form solution involving $X_*$, $S$, $\sigma$, and $\tau$. [Ref. 2] [Ref. 8]. It can also be demonstrated that Bayesian Linear Regression (BLR) is equivalent to Gaussian Process Regression (GPR) with a kernel defined by the dot product.

A Gaussian process $f(x)$ is completely specified by its mean function $m(x)$ and its covariance function $k(x, x')$, where $x \in X$ denotes a point in the index set $X$. These functions are defined as follows:

\begin{equation} m(x) = \mathbb{E}[f(x)] \quad \text{and} \quad k(x, x') = \mathbb{E}[(f(x) - m(x))(f(x') - m(x'))]. \tag{2.6} \end{equation}

As shown in Ref. 8, $f(x)$ as defined in (2.1) constitutes a Gaussian Process because $\theta \sim \mathcal{N}(0, \tau^2 I)$. Furthermore, as shown in this post, $m(x) = 0$ and $k(x, x') = \tau^2 x^T x + \sigma^2$.

Kernel Ridge Regression

The Bayesian linear model suffers from limited expressiveness. A very simple idea to overcome this problem is to project the inputs into some high dimensional space using a set of basis feature space functions, e.g. $\left\{ 1,x,x^{2},\ldots,x^{d-1}\right\} $, and then apply the linear model in this space instead of directly on the inputs themselves. This type of map also defines a simple example of a Gaussian process:

\begin{equation} f(\boldsymbol{x})=\phi\left(\boldsymbol{x}\right)^{T}\boldsymbol{\theta} \tag{2.7}\end{equation}

where $\boldsymbol{x}$ is a $p$ variable input, $\phi\left(\boldsymbol{x}\right)$ is a $p\times d$ dimension vector, and the prior $\boldsymbol{\theta}\sim\mathcal{N}\left(\boldsymbol{0},\Sigma_{p}\right)$, where $\Sigma_{p}$ is a a $p\times p$ covariance matrix, which implies

\begin{align} \mathbb{E}[f(\boldsymbol{x})] & =\phi\left(\boldsymbol{x}\right)^{T}\mathbb{E}[\boldsymbol{\theta}]=0\tag{2.8} \end{align}

\begin{align} \mathbb{E}[f(\boldsymbol{x})f(\boldsymbol{x}')] & =\phi\left(\boldsymbol{x}\right)^{T}\mathbb{E}[\boldsymbol{\theta}\boldsymbol{\theta}^{T}]\phi\left(\boldsymbol{x}'\right)=\phi\left(\boldsymbol{x}\right)^{T}\Sigma_{p}\phi\left(\boldsymbol{x}'\right).\tag{2.9} \end{align}

$f(\boldsymbol{x})$ and $f(\boldsymbol{x}')$ are jointly Gaussian with zero mean and covariance $k(\boldsymbol{x},\boldsymbol{x}')=\phi\left(\boldsymbol{x}\right)^{T}\Sigma_{p}\phi\left(\boldsymbol{x}'\right)$. Therefore any combination of $f(\boldsymbol{x}^{(i)})$, with $\boldsymbol{x}^{(i)}$ in the input set $X$ is a Gaussian process. Using the kernel trick it is possible to show that, fundamentally all we need to know about the feature map $\phi\left(\boldsymbol{x}\right)$ is encapsulated in the corresponding kernel function $k(\boldsymbol{x},\boldsymbol{x}')\ (=\Sigma_{p})$. We only need to specify weighted inner products of the feature space and never need to specify the feature space.

Gaussian Process Regression

Gaussian Process Regression (GPR) excels in dealing with high-dimensional, small-sample nonlinear problems. GPR uses a kernel to define the covariance of a prior distribution over the target functions and leverages observed training data to establish a likelihood function. Employing Bayes' theorem, GPR derives a Gaussian posterior distribution over target functions, utilizing the mean for prediction.

Consider a training set $S = \{(x^{(i)}, y^{(i)})\}_{i=1}^m$ of i.i.d. examples from an unknown distribution. In a noisy regression algorithm, the task is to estimate a function $f$ such that:

\begin{equation} y^{(i)} = f(x^{(i)}) + \varepsilon, \quad i = 1, \ldots, m \tag{3.1} \end{equation}

where $\varepsilon$ are i.i.d. noise terms following a $\mathcal{N}(0, \sigma^2)$ distribution.

We assume a zero-mean Gaussian process prior distribution over functions $f(\cdot)$:

\begin{equation} f \sim \mathcal{GP}(0, k), \tag{3.2} \end{equation}

with a valid covariance function (kernel) $k(\cdot, \cdot)$. An example of kernel is give by the squared-exponential covariance function, given by:

\begin{equation} k(\boldsymbol{x}, \boldsymbol{x}') = \sigma_f \exp\left(-\frac{1}{2l^2} \|\boldsymbol{x} - \boldsymbol{x}'\|^2\right), \tag{3.3} \end{equation}

where $\sigma_f$ and $l$ are hyperparameters. This kernel can be show to be a Bayesian linear regression model with an infinitely dimensional basis [Ref 1].

Now, let $T = \{(x_*^{(i)}, y_*^{(i)})\}_{i=1}^m$ be a test set of i.i.d. examples from the same unknown distribution as $S$. We aim to compute the posterior predictive distribution over the testing outputs $f_*^{(i)}$, associated with $y_*^{(i)}$. For the Gaussian process $f$, the joint distribution of the sets $\{f(x) : x \in X\}$ and $\{f_*(x_*) : x_* \in X_*\}$ is expressed as:

\begin{equation} \left[\begin{array}{c} \boldsymbol{f} \\ \boldsymbol{f_*} \end{array}\right] \bigg| X, X_* \sim \mathcal{N}\left(0, \left[\begin{array}{cc} K(X, X) & K(X, X_*) \\ K(X_*, X) & K(X_*, X_*) \end{array}\right]\right) \Rightarrow \boldsymbol{f_*} | \boldsymbol{f} \sim \text{Normal distribution}. \tag{3.4} \end{equation}

By conditioning a joint Gaussian distribution, we find an expression for $\boldsymbol{f_*} | \boldsymbol{f}$, as follows

\begin{align} \left[\begin{array}{c} \boldsymbol{f} \\ \boldsymbol{f_*} \end{array}\right] \bigg| X, X_* & \sim \mathcal{N}\left(0, \left[\begin{array}{cc} K(X, X) + \sigma^2 I & K(X, X_*) \\ K(X_*, X) & K(X_*, X_*) \end{array}\right]\right). \tag{3.5} \end{align}

If we now consider that we only have access to the noisy version of $f$, i.e., to observations of $y$, the posterior predictive distribution is determined in a similar way as follows [Ref. 1, eq. 2.20-2.24]:

\begin{align} \left[\begin{array}{c} \boldsymbol{y} \\ \boldsymbol{f_*} \end{array}\right] \bigg| X, X_* & = \left[\begin{array}{c} \boldsymbol{f} \\ \boldsymbol{f_*} \end{array}\right] + \left[\begin{array}{c} \boldsymbol{\varepsilon} \\ \boldsymbol{0} \end{array}\right] \sim \mathcal{N}\left(0, \left[\begin{array}{cc} K(X, X) + \sigma^2 I & K(X, X_*) \\ K(X_*, X) & K(X_*, X_*) \end{array}\right]\right). \tag{3.6} \end{align}

The predictive distribution for the test outputs is given by:

\begin{equation} \boxed{\boldsymbol{f_*} | \boldsymbol{y}, X, X_* \sim \mathcal{N}(\mu_*, \Sigma_*)} \tag{3.7} \end{equation}

where $\mu_*$ and $\Sigma_*$ are functions of $X_*$, $S$, $k(\cdot, \cdot)$, and $\sigma$:

\begin{equation} \mu_* = K(X_*, X) \varLambda^{-1} \boldsymbol{y}, \quad \Sigma_* = K(X_*, X_*) - K(X_*, X) \varLambda^{-1} K(X, X_*). \tag{3.8} \end{equation}

where $\varLambda = K(X, X) + \sigma^2 I$.

To find equation (3.7) we used the theorem of the marginal and conditional distributions of the multivariate normal distribution, as shown in Appendices A of [Ref. 1] and [Ref. 3]. However, this is attributed to von Mises [1964, sec. 9.3]. This "conditioning" step is a Bayesian update, where we revise our prior beliefs (GP prior) based on the observed data to compute the posterior predictive distribution.

The particular case of this derivation when $\sigma=0$ corresponds to noise-free observations, where $S = \{(x^{(i)}, f^{(i)})\}_{i=1}^m$ and $T = \{(x_*^{(i)}, f_*^{(i)})\}_{i=1}^m$. In PyMC the noise-free case is modeled with pymc.gp.Latent while the noisy case is modeled with pymc.gp.Marginal.

Numerical implementation of Gaussian Process Regression

Below, we outline the steps of the Gaussian Process Regression algorithm as implemented in scikit-learn's GaussianProcessRegressor and, up to the prior specification for $f$ (using Cholesky decomposition), in PyMC's Gaussian Processes. The procedure follows Algorithm 2.1 from Ref. 1:

Inputs:

• $X$: inputs
• $y$: targets
• $k$: covariance function
• $\sigma^2$: noise level
• $x_*$: single test input

Algorithm Steps:

1. Compute the Cholesky decomposition of $(K + \sigma^2 I)$: $$ L = \text{cholesky}(K(X,X) + \sigma^2 I) $$

   Which means that $L$ is a lower triangular matrix such that $LL^T = K(X,X) + \sigma^2 I$.

2. Solve for $\alpha$: $$ \alpha = L^T \backslash (L \backslash y) $$

   This notation means that we first solve $L \backslash y$, which means find $x: Lx=y$, and then solve $L^T \backslash (L \backslash y)$, which means find $\alpha = (K(X,X)+ \sigma^2 I)^{-1}y$.

3. Compute the predictive mean at the test points: $$ \mu_* = k^T_* \alpha $$

4. Compute the vector of covariances between the test points and the training points: $$ v = L \backslash k_* $$

   where $k_* = K(X, X_*)$ when $X_*$ has a single data point and denotes the vector of covariances between the test point and the $n$ training points. This means that if $X_*$ has multiple data points, we need to compute $k_*$ for each test and training points.

5. Compute the predictive variance at the test points: $$ V[f_*] = k(x_*, x_*) - v^T v $$

   where $V[f_*] = \Sigma_*$ when $X_*$ has a single test data point.

6. Calculate the log marginal likelihood: $$ \log p(y|X) = -\frac{1}{2} y^T \alpha - \sum \log L_{ii} - \frac{n}{2} \log 2\pi $$

Return:

• $\mu_*$: predictive mean
• $\Sigma_*$: predictive variance
• $\log p(y|X)$: log marginal likelihood

Notes:

• For multiple test cases, repeat calculations of $\mu_*$ and $V[f_*]=\Sigma_*$.
• These quantities, when plugged in the distribution $\boldsymbol{f_*} | \boldsymbol{y}, X, X_* \sim \mathcal{N}(\mu_*, \Sigma_*)$, represent the updated beliefs about the function at the test point $x_*$ after observing the train data.
• Computational complexity is $n^3/6$ for the Cholesky decomposition and $n^2/2$ for solving triangular systems.

Model Selection

The Gaussian Process Regressor is useful if we have enough prior information about a dataset to confidently specify prior mean and covariance functions. However, having such detailed prior information is not typically the case in applications. We may use a hierarchical specification to represent vague prior information simply in the form of $m$ and $k$, and optimize these introduced hyperparameters based on the data.

Maximum Likelihood Estimation

In libraries such as scikit-learn and GPy, model selection is implemented using a frequentist approach commonly employed in GPR model selection via Maximum Likelihood Estimation [Ref. 3]. We now introduce this method. We start by defining a model:

\begin{equation} f \sim \mathcal{GP}(m, k), \tag{4.1} \end{equation}

where the Squared Exponential (SE) Kernel, including an additional noise term, is given by:

\begin{align} m(\boldsymbol{x}) &= a\boldsymbol{x}^2 + b\boldsymbol{x} + c, \tag{4.2} \end{align}

\begin{align} k(\boldsymbol{x}, \boldsymbol{x}') &= \sigma_f^2 \exp\left(-\frac{1}{2}(\boldsymbol{x} - \boldsymbol{x}')^T M (\boldsymbol{x} - \boldsymbol{x}')\right) + \sigma_n^2 \delta_{ii'}. \tag{4.3} \end{align}

We define the set of hyperparameters as:

\begin{equation} \Theta = \{a, b, c, \{M\}, \sigma_f, \sigma_n\}, \tag{4.4} \end{equation}

where $M$ can be defined in terms of the characteristic length-scales of correlations, $l_i$, which define the length scales in the input space where correlations become negligible. For high-dimensional datasets, this matrix could identify a few directions in the input space with especially high relevance (in the sense of PCA - high variance). Although we use the squared exponential, any positive definite function can be used as a covariance function.

The optimal hyperparameters, denoted as $\Theta^*$, are determined by maximizing the log marginal likelihood. This can be expressed as:

\begin{equation} \Theta^* = \arg \max_\Theta \log p(y | X, \Theta). \tag{4.5} \end{equation}

Bayesian Model Selection

The primary distinction between the Bayesian approach to inference and other optimization-based methods, such as MLE, lies in the marginal likelihood, which involves integrating over the parameter space. PyMC employs a fully Bayesian framework. It treats hyperparameters (or parameters as they are called in Bayesian statistics) as random variables with specified prior distributions (which may depend on other constants called hyperparameter in this context). Inference is performed over the full posterior distribution of both the GP function and its hyperparameters, providing a complete uncertainty quantification. The hierarchical structure for such an approach is completely described in section 5.2 of [Ref. 1], and it follows the general steps of utilizing priors of parameters, hyperparameters, and the likelihood function to sample the posterior distribution. As usual for this library, the posterior distribution is sampled using Markov Chain Monte Carlo (MCMC) methods such as in the Nuts sampler.

Numerical Examples

We will now illustrate the Gaussian Process Regression algorithm using synthetic data. We will use the scikit-learn library to implement the algorithms for the Bayesian-Frequentist approach. We will also demonstrate the pure Bayesian approach using the PyMC library and compare the results.

Simple Linear Function Example

Bayesian-Frequentist mixed approach with scikit-learn

Now we use the DotProduct kernel from sklearn library to show an example of GP in action for a linear function.

As an example of linear function we use

\begin{equation} f(x)=0.4x+10, \tag{5.1} \end{equation}

on the range $x\in[0,100]$ for train data and $x\in[0,150]$ for test data. We generate both train and test data utilizing this function with white noise $\epsilon=\mathcal{N}\left(0,\sigma^{2}\right)$, where $\sigma=1$.

import numpy as np
from sklearn.metrics import mean_squared_error
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import ConstantKernel, DotProduct
from matplotlib import pyplot as plt
import seaborn as sns

import arviz as az
%config InlineBackend.figure_format = 'retina'
az.style.use("arviz-darkgrid")

# Set random seed for reproducibility
np.random.seed(42)


# Define the linear function
def func(x):
    return 0.4 * x + 10


# Generate a random training dataset with seed
def generate_random_train_dataset(func, x_end, num_train, sigma_noise, seed=42):
    np.random.seed(seed)
    x = np.random.uniform(0, x_end, num_train)  # Randomly sample x values
    noise = np.random.normal(0, sigma_noise, num_train)  # Add Gaussian noise
    y = func(x) + noise
    return x, y


# Generate a test dataset with evenly spaced x values
def generate_random_test_dataset(func, x_end, num_test, sigma_noise, seed=42):
    np.random.seed(seed)
    x = np.linspace(0, x_end, num_test)  # Generate evenly spaced x values
    noise = np.random.normal(0, sigma_noise, num_test)  # Add Gaussian noise
    y = func(x) + noise
    return x, y


def gaussian_process_regressor(x, x_train, y_train, x_test, y_test, kernel, sigma_n=2):
    # Initialize Gaussian Process Regressor with the given kernel and noise level
    gp = GaussianProcessRegressor(kernel=kernel, alpha=sigma_n**2, n_restarts_optimizer=30)

    # Fit the GPR model on the training data
    gp.fit(x_train.reshape(-1, 1), y_train)

    # Predict mean and standard deviation on the test data
    y_prediction, std = gp.predict(x_test.reshape(-1, 1), return_std=True)

    # Evaluate model performance using MSE
    mse = mean_squared_error(y_test, y_prediction)
    print(f"MSE of Test Set: {mse:.4f}")

    # Display fitted kernel parameters
    params = gp.kernel_.get_params()
    print("Fitted Kernel Parameters:")
    display(params)

    # Generate samples from the posterior distribution
    y_posterior_samples = gp.sample_y(x.reshape(-1, 1), n_samples=1)

    # Compute mean and standard deviation of the posterior samples
    y_posterior = np.mean(y_posterior_samples, axis=1)

    return y_prediction, std, params, y_posterior


def plot_gp_predictions(x, x_test, y_test, x_train, y_train, f, y_prediction, y_posterior, std):
    # Sort x_test and corresponding predictions for smooth plotting
    sort_index = np.argsort(x_test)

    # Create the plot
    fig, ax = plt.subplots(figsize=(12, 5))

    # Scatter plot of test data
    sns.scatterplot(x=x_test, y=y_test, label="Test Data", ax=ax, color="orange")

    # Scatter plot of training data
    sns.scatterplot(x=x_train, y=y_train, label="Train Data", marker="P", s=150, ax=ax, color="black")

    # Plot the ground truth function
    sns.lineplot(x=x_test, y=f(x_test), color="red", label="True Function - f(x_test)", ax=ax)

    # Plot the credible interval
    ax.fill_between(x=x_test[sort_index], y1=(y_prediction - 2 * std)[sort_index], y2=(y_prediction + 2 * std)[sort_index], color="green", alpha=0.25, label="Credible Interval (±2σ)")

    # Plot the predicted mean
    sns.lineplot(x=x_test, y=y_prediction, color="green", label="Prediction (Mean)", ax=ax)

    # Plot the posterior samples
    sns.lineplot(x=x, y=y_posterior, color="blue", linestyle="-.", label="Posterior Sample Example", ax=ax)

    # Customize plot appearance
    ax.set_title("Gaussian Process Regression with Maximum Likelihood Optimization on Hyperparameters", fontsize=16)
    ax.set_xlabel("x", fontsize=16)
    ax.set_ylabel("y", fontsize=16)
    ax.grid(True)
    ax.legend(loc="upper left", fontsize=12)

    # Display the plot
    plt.show()

# Standard deviation of noise
sigma_noise = 1.0

# Generate synthetic data

x_values = np.linspace(0, 150, 1000)  # Range of x values for evaluation
x_train, y_train = generate_random_train_dataset(func, x_end=100, num_train=3, sigma_noise=sigma_noise)  # Training dataset
x_test, y_test = generate_random_test_dataset(func, x_end=150, num_test=200, sigma_noise=sigma_noise)  # Test dataset

# Define kernel parameters
sigma_f = 1.0  # Scale of the kernel function
sigma_0 = 100.0  # Noise level parameter for the DotProduct kernel

# Define the kernel
kernel_dot = ConstantKernel(constant_value=sigma_f) * DotProduct(sigma_0=sigma_0)

# Apply Gaussian Process Regressor
y_prediction, std, params, y_posterior = gaussian_process_regressor(x=x_values, x_train=x_train, y_train=y_train, x_test=x_test, y_test=y_test, kernel=kernel_dot)

# Plot the results
plot_gp_predictions(x=x_values, x_test=x_test, y_test=y_test, x_train=x_train, y_train=y_train, f=func, y_prediction=y_prediction, y_posterior=y_posterior, std=std)

MSE of Test Set: 3.6980
Fitted Kernel Parameters:

{'k1': 0.442**2,
 'k2': DotProduct(sigma_0=17.2),
 'k1__constant_value': 0.1957795723545119,
 'k1__constant_value_bounds': (1e-05, 100000.0),
 'k2__sigma_0': 17.19854740387973,
 'k2__sigma_0_bounds': (1e-05, 100000.0)}

Pure Bayesian approach with PyMC

We begin by exploring the relationship between the DotProduct kernel from scikit-learn and the Linear kernel from PyMC to understand linear data relationships. This comparison helps in distinguishing between the mixed Bayesian-frequentist and fully Bayesian approaches described above.

The DotProduct kernel in scikit-learn is defined as:

\begin{equation} k(x, x') = \tau^2 x^T x' + \sigma^2, \tag{6.1} \end{equation}

where $\tau^2$ scales the linear function and $\sigma^2$ represents the noise level.

To implement a similar kernel in PyMC, we need to adjust the standard Linear kernel, which is defined as:

\begin{equation} k(x, x') = (x - c)(x' - c), \tag{6.3} \end{equation}

Here, $c$ represents a constant offset.

For a direct comparison, the Linear kernel in PyMC can be modified to match the form of the DotProduct kernel in scikit-learn by setting $c = 0$ and incorporating a noise term, resulting in the following expression:

\begin{equation} \text{DotProduct}_{\text{PyMC}} = \text{Linear}(c=0) + \text{Constant}(\sigma^2), \tag{6.4} \end{equation}

This adaptation allows us to effectively compare the two approaches under similar conditions.

import pymc as pm

with pm.Model() as model:
    # Priors for kernel parameters - same as the ones used in the previous example
    sigma_f = pm.HalfNormal("sigma_f", sigma=10)  # Scale parameter
    sigma_0 = pm.HalfNormal("sigma_0", sigma=100)  # Noise parameter for Dot Product kernel

    # sigma_f = pm.Uniform("sigma_f", lower=0, upper=10)  # Scale parameter
    # sigma_0 = pm.Uniform("sigma_0", lower=0, upper=100)  # Noise parameter for Dot Product kernel

    # Define the kernel
    cov = pm.gp.cov.Constant(sigma_f**2) * (pm.gp.cov.Linear(input_dim=1, c=0) + pm.gp.cov.Constant(sigma_0**2))

    gp = pm.gp.Marginal(cov_func=cov)

    # sigma = pm.Deterministic("sigma", sigma_noise)
    y_ = gp.marginal_likelihood("y", X=x_train.reshape(-1, 1), y=y_train, sigma=sigma_noise)

    marginal_post = pm.sample(5000, tune=2000, target_accept=0.95, nuts_sampler="numpyro")

    # add the GP conditional to the model, given the new X values
    f_pred = gp.conditional("f_pred", x_test.reshape(-1, 1))

    # Predict y values at test points using the posterior samples
    y_pred = gp.conditional("y_pred", x_test.reshape(-1, 1), pred_noise=True)
    y_samples = pm.sample_posterior_predictive(marginal_post.sel(draw=slice(0, 20)), var_names=["y_pred"])

    pred_samples = pm.sample_posterior_predictive(marginal_post.sel(draw=slice(0, 20)), var_names=["f_pred"])

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Sampling: [y_pred]

Output()

Sampling: [f_pred]

Output()

import arviz as az

summary = az.summary(marginal_post, var_names=["sigma_f", "sigma_0"], round_to=2, kind="stats")
summary

	mean	sd	hdi_3%	hdi_97%
sigma_f	1.70	2.30	0.14	5.73
sigma_0	45.56	43.89	0.39	128.40

# plot the results
fig = plt.figure(figsize=(12, 5))
ax = fig.gca()

# plot the samples from the gp posterior with samples and shading
from pymc.gp.util import plot_gp_dist

f_pred_samples = az.extract(pred_samples, group="posterior_predictive", var_names=["f_pred"])
plot_gp_dist(ax, samples=f_pred_samples.T, x=x_test, fill_alpha=0.6, samples_alpha=0.1)

# # Add a label for the posterior predictive
plt.plot([], [], color="red", label=r"Posterior Samples of $f(x)$")

# plot the data and the true latent function
plt.plot(x_values, func(x_values), "dodgerblue", lw=3, label="True f")
plt.scatter(x_train, y_train, marker="P", s=150, color="black", label="Observed data")

# axis labels and title
plt.xlabel("X")
plt.title("Posterior distribution over $f(x)$ at the observed values")
plt.legend()  # This will now include the legend for red posterior predictions as well.

<matplotlib.legend.Legend at 0x7727764af320>

# plot the results
fig = plt.figure(figsize=(12, 5))
ax = fig.gca()

# plot the samples from the gp posterior with samples and shading
from pymc.gp.util import plot_gp_dist

# posterior predictive distribution
y_pred_samples = az.extract(y_samples, group="posterior_predictive", var_names=["y_pred"])
plot_gp_dist(ax, y_pred_samples.T, x=x_test, plot_samples=True, palette="bone_r")

# # Add a label for the posterior predictive
plt.plot([], [], color="black", label=r"Posterior Samples of $y(x)$")

# overlay a scatter of one draw of random points from the
#   posterior predictive distribution
plt.plot(x_test, y_pred_samples[:, 0], "co", ms=2)

# Add a label for the posterior predicted data
plt.scatter([], [], color="lightblue", label="Single Sample of $y(x)$")

# plot the data and the true latent function
plt.plot(x_values, func(x_values), "dodgerblue", lw=3, label="True f")
plt.scatter(x_train, y_train, marker="P", s=150, color="red", label="Observed data")

# axis labels and title
plt.xlabel("X")
plt.title("Posterior distribution over $f(x)$ at the observed values")
plt.legend()  # This will now include the legend for red posterior predictions as well.

<matplotlib.legend.Legend at 0x7727765da4b0>

fig, axs = plt.subplots(1, 2, figsize=(10, 4))
axs = axs.flatten()

# plot sigma_f
az.plot_posterior(marginal_post, var_names=["sigma_f"], ax=axs[0], hdi_prob=0.95)
axs[0].set_title("Posterior of sigma_f")

# plot sigma_0
az.plot_posterior(marginal_post, var_names=["sigma_0"], ax=axs[1], hdi_prob=0.95)
axs[1].set_title("Posterior of sigma_0")

plt.tight_layout()
plt.show()

/tmp/ipykernel_2879542/2061803742.py:12: UserWarning: The figure layout has changed to tight
  plt.tight_layout()

# check Rhat, values above 1 may indicate convergence issues
n_nonconverged = int(np.sum(az.rhat(marginal_post)[["sigma_f", "sigma_0"]].to_array() > 1.03).values)
if n_nonconverged == 0:
    print("No Rhat values above 1.03, \N{check mark}")
else:
    print(f"The MCMC chains for {n_nonconverged} RVs appear not to have converged.")

No Rhat values above 1.03, ✓

We clearly see the advantages of the fully Bayesian approach offered by PyMC, which provides more comprehensive uncertainty quantification. The mixed Bayesian-classical approach in scikit-learn is more straightforward and computationally efficient but lacks the benefits of the full Bayesian framework. For example, PyMC samples the model parameters from the posterior distribution, providing a more complete understanding of the model's uncertainty, whereas scikit-learn provides only point estimates. More importantly, the PyMC approach allows us to obtain samples from the posterior distribution of the function $f$, rather than relying solely on the mean and variance estimates as in the scikit-learn implementation.

The first plot above for the PyMC illustrates that the DotProduct kernel generates only linear samples for $f$, which aligns with the theoretical derivation that a Bayesian Linear Regression (BLR) is equivalent to a Gaussian Process Regression (GPR) with a DotProduct kernel. Notice how the predictive uncertainty increases for input values that are distant from any training points.

Another Example with a More Complex Function

Let us experiment with a more complex, non-linear function:

$$ f(x) = \sin(4\pi x) + \sin(7\pi x), \tag{8.1} $$

defined over the range $x \in [0, 2]$ for training data and $x \in [0, 3]$ for testing data. We generate both training and testing data using this function, adding white noise $\epsilon = \mathcal{N}(0, \sigma^2)$, where $\sigma = 0.1$.

Nonlinear function with ExpSineSquared

Bayesian-Frequentist mixed approach with scikit-learn

\begin{equation} k(x,x')=\sigma_{f}\exp\left(-\frac{2\sin^{2}\left(\pi\Vert x-x'\Vert/p\right)}{l^{2}}\right), \tag{9.1} \end{equation}

This kernel allows modeling periodic functions

#nonlinear function
def g(x):
#     g_x = np.exp(-x) * np.sin(4*x) + (x-1)*np.heaviside(x-1,0) - 0.732
    g_x = np.sin((4*np.pi)*x) + np.sin((7*np.pi)*x)
    return g_x

sigma_noise_data = 0.1

x = np.linspace(start=0, stop=3, num=500)
x_train,y_train = generate_random_train_dataset(g,2,100,sigma_noise_data)
x_test,y_test = generate_random_test_dataset(g,3,300,sigma_noise_data)

from sklearn.gaussian_process.kernels import ExpSineSquared
oscilating_kernel = ConstantKernel(constant_value=3, constant_value_bounds=(1e-2,10)) * ExpSineSquared(length_scale=1, periodicity=1) \
                    + ConstantKernel(constant_value=1, constant_value_bounds=(1e-2, 10)) * ExpSineSquared(length_scale=1, periodicity=1)

sigma_n = 0.5
y_prediction, std, params, y_posterior = gaussian_process_regressor(x, x_train, y_train, x_test, y_test, oscilating_kernel,sigma_n)

plot_gp_predictions(x, x_test, y_test, x_train, y_train, g, y_prediction, y_posterior, std)

MSE of Test Set: 0.4883
Fitted Kernel Parameters:

{'k1': 1.19**2 * ExpSineSquared(length_scale=1.05, periodicity=0.562),
 'k2': 0.516**2 * ExpSineSquared(length_scale=0.000113, periodicity=5.84e+03),
 'k1__k1': 1.19**2,
 'k1__k2': ExpSineSquared(length_scale=1.05, periodicity=0.562),
 'k1__k1__constant_value': 1.4165260761128176,
 'k1__k1__constant_value_bounds': (0.01, 10),
 'k1__k2__length_scale': 1.0531039433006095,
 'k1__k2__periodicity': 0.5621282748466709,
 'k1__k2__length_scale_bounds': (1e-05, 100000.0),
 'k1__k2__periodicity_bounds': (1e-05, 100000.0),
 'k2__k1': 0.516**2,
 'k2__k2': ExpSineSquared(length_scale=0.000113, periodicity=5.84e+03),
 'k2__k1__constant_value': 0.26590326177625423,
 'k2__k1__constant_value_bounds': (0.01, 10),
 'k2__k2__length_scale': 0.00011260432532052758,
 'k2__k2__periodicity': 5840.965885405671,
 'k2__k2__length_scale_bounds': (1e-05, 100000.0),
 'k2__k2__periodicity_bounds': (1e-05, 100000.0)}

Pure Bayesian approach with PyMC

with pm.Model() as model:
    # Define the kernel
    sigma1 = pm.HalfNormal("sigma1", sigma=1)
    # length_scale1 = pm.Gamma("length_scale1", alpha=2, beta=0.5)
    # period1 = pm.Gamma("period1", alpha=2, beta=0.5)
    # length_scale1 = pm.HalfNormal("length_scale1", sigma=1)
    # period1 = pm.HalfNormal("period1", sigma=1)
    length_scale1 = pm.Normal("length_scale1", mu=1, sigma=1)
    period1 = pm.Normal("period1", mu=1, sigma=1)
    mod1 = pm.gp.cov.Constant(sigma1**2) * pm.gp.cov.Periodic(input_dim=1, period= period1, ls=length_scale1)

    sigma2 = pm.HalfNormal("sigma2", sigma=0.5)
    # length_scale2 = pm.Gamma("length_scale2", alpha=2, beta=0.5)
    # period2 = pm.Gamma("period2", alpha=2, beta=0.5)
    length_scale2 = pm.HalfNormal("length_scale2", sigma=1)
    period2 = pm.HalfNormal("period2", sigma=100)
    # length_scale2 = pm.Normal("length_scale2", mu=1, sigma=1)
    # period2 =  pm.Normal("period2", mu=1, sigma=1)
    mod2 = pm.gp.cov.Constant(sigma2**2) * pm.gp.cov.Periodic(input_dim=1, period= period2, ls=length_scale2)

    cov = mod1 + mod2

    gp = pm.gp.Marginal(cov_func=cov)

    # sigma = pm.Deterministic("sigma", sigma_noise)
    y_ = gp.marginal_likelihood("y", X=x_train.reshape(-1, 1), y=y_train, sigma=sigma_noise)

    marginal_post = pm.sample(5000, tune=1000, target_accept=0.98, nuts_sampler="numpyro")

    # add the GP conditional to the model, given the new X values
    f_pred = gp.conditional("f_pred", x_test.reshape(-1, 1))

    # Predict y values at test points using the posterior samples
    y_pred = gp.conditional("y_pred", x_test.reshape(-1, 1), pred_noise=True)
    y_samples = pm.sample_posterior_predictive(marginal_post.sel(draw=slice(0, 20)), var_names=["y_pred"])

    pred_samples = pm.sample_posterior_predictive(marginal_post.sel(draw=slice(0, 20)), var_names=["f_pred"])

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There were 16 divergences after tuning. Increase `target_accept` or reparameterize.
The rhat statistic is larger than 1.01 for some parameters. This indicates problems during sampling. See https://arxiv.org/abs/1903.08008 for details
The effective sample size per chain is smaller than 100 for some parameters.  A higher number is needed for reliable rhat and ess computation. See https://arxiv.org/abs/1903.08008 for details
Sampling: [y_pred]

Output()

Sampling: [f_pred]

Output()

summary = az.summary(marginal_post, var_names=["sigma1", "length_scale1", "period1", "sigma2", "length_scale2", "period2"], round_to=2, kind="stats")
summary

	mean	sd	hdi_3%	hdi_97%
sigma1	1.20	0.45	0.48	2.04
length_scale1	0.50	0.20	0.17	0.86
period1	0.70	0.23	0.53	1.07
sigma2	0.47	0.33	0.00	1.03
length_scale2	0.80	0.55	0.00	1.76
period2	59.17	62.03	0.26	175.85

# plot the results
fig = plt.figure(figsize=(12, 5))
ax = fig.gca()

# plot the samples from the gp posterior with samples and shading
from pymc.gp.util import plot_gp_dist

f_pred_samples = az.extract(pred_samples, group="posterior_predictive", var_names=["f_pred"])
plot_gp_dist(ax, samples=f_pred_samples.T, x=x_test, fill_alpha=0.6, samples_alpha=0.1)

# # Add a label for the posterior predictive
plt.plot([], [], color="red", label=r"Posterior Samples of $f(x)$")

# plot the data and the true latent function
plt.plot(x_test, g(x_test), "dodgerblue", lw=3, label="True f")
plt.scatter(x_train, y_train, marker="P", s=150, color="black", label="Observed data")

# axis labels and title
plt.xlabel("X")
plt.title("Posterior distribution over $f(x)$ at the observed values")
plt.legend()  # This will now include the legend for red posterior predictions as well.

<matplotlib.legend.Legend at 0x77276bff8410>

import matplotlib.pyplot as plt
import arviz as az

# Create a larger grid for the subplots
fig, axs = plt.subplots(2, 3, figsize=(20, 8))  # Adjusted for 6 plots
axs = axs.flatten()

# Plot sigma1
az.plot_posterior(marginal_post, var_names=["sigma1"], ax=axs[0], hdi_prob=0.95)
axs[0].set_title("Posterior of sigma1")

# Plot length_scale1
az.plot_posterior(marginal_post, var_names=["length_scale1"], ax=axs[1], hdi_prob=0.95)
axs[1].set_title("Posterior of length_scale1")

# Plot period1
az.plot_posterior(marginal_post, var_names=["period1"], ax=axs[2], hdi_prob=0.95)
axs[2].set_title("Posterior of period1")

# Plot sigma2
az.plot_posterior(marginal_post, var_names=["sigma2"], ax=axs[3], hdi_prob=0.95)
axs[3].set_title("Posterior of sigma2")

# Plot length_scale2
az.plot_posterior(marginal_post, var_names=["length_scale2"], ax=axs[4], hdi_prob=0.95)
axs[4].set_title("Posterior of length_scale2")

# Plot period2
az.plot_posterior(marginal_post, var_names=["period2"], ax=axs[5], hdi_prob=0.95)
axs[5].set_title("Posterior of period2")

plt.tight_layout()
plt.show()

/tmp/ipykernel_2879542/1562436338.py:32: UserWarning: The figure layout has changed to tight
  plt.tight_layout()

# check Rhat, values above 1 may indicate convergence issues
n_nonconverged = int(np.sum(az.rhat(marginal_post)[["sigma1", "length_scale1", "period1", "sigma2", "length_scale2", "period2"]].to_array() > 1.03).values)
if n_nonconverged == 0:
    print("No Rhat values above 1.03, \N{check mark}")
else:
    print(f"The MCMC chains for {n_nonconverged} RVs appear not to have converged.")

The MCMC chains for 3 RVs appear not to have converged.

Gaussian Process Regression (GPR) on Mauna Loa CO2 Data

Now we apply GPR to a real-world dataset. We analyze the CO2 atmospheric readings from the Mauna Loa Observatory. This example is inspired by Section 5.4.3 of "Gaussian Processes for Machine Learning" by Rasmussen & Williams. It demonstrates complex kernel engineering and hyperparameter optimization based on the log-marginal-likelihood. The dataset includes monthly average atmospheric CO2 concentrations (in parts per million by volume (ppm)) collected at the Mauna Loa Observatory in Hawaii from 1958 to 2001. The objective is to model the CO2 concentration as a function of time and to extrapolate for years after 2001. Additional details and the dataset can be found here.

Bayesian-Frequentist mixed approach with scikit-learn

from sklearn.datasets import fetch_openml
import pandas as pd
from sklearn.gaussian_process.kernels import RBF, ExpSineSquared, RationalQuadratic, WhiteKernel
import datetime

# Fetch the CO2 dataset from OpenML
co2 = fetch_openml(data_id=41187, as_frame=True)

# Prepare the data
co2_data = co2.frame.copy()
co2_data["date"] = pd.to_datetime(co2_data[["year", "month", "day"]])
co2_data.set_index("date", inplace=True)
co2_data = co2_data[["co2"]].dropna()

# Resample to monthly averages
monthly_co2 = co2_data.resample("ME").mean().dropna()

# Prepare data for Gaussian Process Regression
X = (monthly_co2.index.year + monthly_co2.index.month / 12).to_numpy().reshape(-1, 1)
y = monthly_co2["co2"].to_numpy()

# Define the kernel components
long_term_trend_kernel = 50.0**2 * RBF(length_scale=50.0)
seasonal_kernel = 2.0**2 * RBF(length_scale=100.0) * ExpSineSquared(length_scale=1.0, periodicity=1.0, periodicity_bounds="fixed")
irregularities_kernel = 0.5**2 * RationalQuadratic(length_scale=1.0, alpha=1.0)
noise_kernel = 0.1**2 * RBF(length_scale=0.1) + WhiteKernel(noise_level=0.1**2, noise_level_bounds=(1e-5, 1e5))

# Combine kernels
co2_kernel = long_term_trend_kernel + seasonal_kernel + irregularities_kernel + noise_kernel

# Initialize and fit the Gaussian Process model
y_mean = y.mean()
gaussian_process = GaussianProcessRegressor(kernel=co2_kernel, normalize_y=False)
gaussian_process.fit(X, y - y_mean)

# Display fitted kernel parameters
params = gaussian_process.kernel_.get_params()
print("Fitted Kernel Parameters:")
display(params)

# Generate predictions
today = datetime.datetime.now()
current_month = today.year + today.month / 12
X_test = np.linspace(1958, current_month, num=1000).reshape(-1, 1)
mean_y_pred, std_y_pred = gaussian_process.predict(X_test, return_std=True)
mean_y_pred += y_mean

# Plot raw data and Gaussian Process results together
plt.figure(figsize=(16, 9))

# Plot daily measurements
plt.plot(co2_data.index, co2_data["co2"], color="gray", linestyle="-", label="Daily Measurements")

# Plot monthly averages used for modeling
plt.plot(monthly_co2.index, monthly_co2["co2"], color="black", linestyle="dashed", label="Monthly Averages - Train")

# Plot Gaussian Process Prediction
plt.plot(pd.to_datetime((X_test.flatten() - 1958) * 365.25, unit="D", origin="1958-01-01"), mean_y_pred, color="tab:blue", alpha=0.8, label="Gaussian Process Prediction")

# Plot confidence interval
plt.fill_between(pd.to_datetime((X_test.flatten() - 1958) * 365.25, unit="D", origin="1958-01-01"), mean_y_pred - std_y_pred, mean_y_pred + std_y_pred, color="tab:blue", alpha=0.2)

plt.legend()
plt.xlabel("Year")
plt.ylabel("CO$_2$ Concentration (ppm)")
plt.title("CO$_2$ Concentration Measurements and Gaussian Process Prediction\nfrom the Mauna Loa Observatory")
plt.show()

Fitted Kernel Parameters:

{'k1': 44.8**2 * RBF(length_scale=51.6) + 2.64**2 * RBF(length_scale=91.5) * ExpSineSquared(length_scale=1.48, periodicity=1) + 0.536**2 * RationalQuadratic(alpha=2.89, length_scale=0.968),
 'k2': 0.188**2 * RBF(length_scale=0.122) + WhiteKernel(noise_level=0.0367),
 'k1__k1': 44.8**2 * RBF(length_scale=51.6) + 2.64**2 * RBF(length_scale=91.5) * ExpSineSquared(length_scale=1.48, periodicity=1),
 'k1__k2': 0.536**2 * RationalQuadratic(alpha=2.89, length_scale=0.968),
 'k1__k1__k1': 44.8**2 * RBF(length_scale=51.6),
 'k1__k1__k2': 2.64**2 * RBF(length_scale=91.5) * ExpSineSquared(length_scale=1.48, periodicity=1),
 'k1__k1__k1__k1': 44.8**2,
 'k1__k1__k1__k2': RBF(length_scale=51.6),
 'k1__k1__k1__k1__constant_value': 2005.421371197315,
 'k1__k1__k1__k1__constant_value_bounds': (1e-05, 100000.0),
 'k1__k1__k1__k2__length_scale': 51.595329572387215,
 'k1__k1__k1__k2__length_scale_bounds': (1e-05, 100000.0),
 'k1__k1__k2__k1': 2.64**2 * RBF(length_scale=91.5),
 'k1__k1__k2__k2': ExpSineSquared(length_scale=1.48, periodicity=1),
 'k1__k1__k2__k1__k1': 2.64**2,
 'k1__k1__k2__k1__k2': RBF(length_scale=91.5),
 'k1__k1__k2__k1__k1__constant_value': 6.978134208694815,
 'k1__k1__k2__k1__k1__constant_value_bounds': (1e-05, 100000.0),
 'k1__k1__k2__k1__k2__length_scale': 91.47857462300647,
 'k1__k1__k2__k1__k2__length_scale_bounds': (1e-05, 100000.0),
 'k1__k1__k2__k2__length_scale': 1.4846552902943533,
 'k1__k1__k2__k2__periodicity': 1.0,
 'k1__k1__k2__k2__length_scale_bounds': (1e-05, 100000.0),
 'k1__k1__k2__k2__periodicity_bounds': 'fixed',
 'k1__k2__k1': 0.536**2,
 'k1__k2__k2': RationalQuadratic(alpha=2.89, length_scale=0.968),
 'k1__k2__k1__constant_value': 0.28764505470696,
 'k1__k2__k1__constant_value_bounds': (1e-05, 100000.0),
 'k1__k2__k2__length_scale': 0.9678376308386418,
 'k1__k2__k2__alpha': 2.885188936804739,
 'k1__k2__k2__length_scale_bounds': (1e-05, 100000.0),
 'k1__k2__k2__alpha_bounds': (1e-05, 100000.0),
 'k2__k1': 0.188**2 * RBF(length_scale=0.122),
 'k2__k2': WhiteKernel(noise_level=0.0367),
 'k2__k1__k1': 0.188**2,
 'k2__k1__k2': RBF(length_scale=0.122),
 'k2__k1__k1__constant_value': 0.03547936805029443,
 'k2__k1__k1__constant_value_bounds': (1e-05, 100000.0),
 'k2__k1__k2__length_scale': 0.12165629885448888,
 'k2__k1__k2__length_scale_bounds': (1e-05, 100000.0),
 'k2__k2__noise_level': 0.03665940202133469,
 'k2__k2__noise_level_bounds': (1e-05, 100000.0)}

The results above demonstrate hability of the GPR to learn from the data and make predictions. The model is able to capture the trend in the data and provide reasonable predictions for the future. Now we will discuss the Bayesian approach with PyMC.

Fully Bayesian Approach with PyMC

We implemented two versions of the GPR kernel for the Mauna Loa CO2 dataset in PyMC. Unfortunately, neither of them worked, as the sampling time was excessively long. The difference between the kernels lies in their complexity; in the simpler version, we fixed a few parameters, while in the more complex version, we sampled all the parameters. The values of the fixed parameters were inspired by the results from the mixed Bayesian-Frequentist approach discussed above. The kernels are defined as follows:

# Define the Gaussian Process kernel
with pm.Model() as model_complex:
    # Long-term trend kernel
    sigma_lt = pm.HalfNormal("sigma_lt", sigma=50.0)
    ls_lt = pm.Gamma("ls_lt", alpha=2, beta=0.04)
    long_term_trend_kernel = pm.gp.cov.Constant(sigma_lt**2) * pm.gp.cov.ExpQuad(input_dim=1, ls=ls_lt)

    # Seasonal kernel
    sigma_seasonal = pm.HalfNormal("sigma_seasonal", sigma=2.0)
    ls_seasonal = pm.Gamma("ls_seasonal", alpha=2, beta=0.02)
    ls_periodic = pm.Gamma("ls_periodic", alpha=2, beta=2.0)
    period = 1.0  # Fixed
    seasonal_kernel = pm.gp.cov.Constant(sigma_seasonal**2) * pm.gp.cov.ExpQuad(input_dim=1, ls=ls_seasonal) * pm.gp.cov.Periodic(input_dim=1, period=period, ls=ls_periodic)

    # Irregularities kernel
    sigma_irregular = pm.HalfNormal("sigma_irregular", sigma=0.5)
    ls_irregular = pm.Gamma("ls_irregular", alpha=2, beta=2.0)
    alpha_irregular = pm.Gamma("alpha_irregular", alpha=2, beta=1.0)
    irregularities_kernel = pm.gp.cov.Constant(sigma_irregular**2) * pm.gp.cov.RatQuad(input_dim=1, ls=ls_irregular, alpha=alpha_irregular)

    # Noise kernel
    sigma_noise_c = pm.HalfNormal("sigma_noise_c", sigma=0.12)
    ls_noise = pm.Gamma("ls_noise", alpha=2, beta=20.0)
    sigma_noise_w = pm.HalfNormal("sigma_noise_w", sigma=0.12)
    noise_kernel = pm.gp.cov.Constant(sigma_noise_c) * pm.gp.cov.ExpQuad(input_dim=1, ls=ls_noise) + pm.gp.cov.WhiteNoise(sigma=sigma_noise_w)

    # Combine the kernels
    co2_kernel = long_term_trend_kernel + seasonal_kernel + irregularities_kernel + noise_kernel

# Define the Gaussian Process kernel
with pm.Model() as model_simpler:
    # Long-term trend kernel
    ls_lt = pm.Gamma("ls_lt", alpha=2, beta=0.04)
    long_term_trend_kernel = 44.8**2 * pm.gp.cov.ExpQuad(input_dim=1, ls=ls_lt)

    # Seasonal kernel
    ls_seasonal = pm.Gamma("ls_seasonal", alpha=2, beta=0.02)
    ls_periodic = pm.Gamma("ls_periodic", alpha=2, beta=2.0)
    period = 1.0  # Fixed
    seasonal_kernel = 2.64**2 * pm.gp.cov.ExpQuad(input_dim=1, ls=ls_seasonal) * pm.gp.cov.Periodic(input_dim=1, period=period, ls=ls_periodic)

    # Irregularities kernel
    ls_irregular = pm.Gamma("ls_irregular", alpha=2, beta=2.0)
    alpha_irregular = pm.Gamma("alpha_irregular", alpha=2, beta=1.0)
    irregularities_kernel = 0.536**2  * pm.gp.cov.RatQuad(input_dim=1, ls=ls_irregular, alpha=alpha_irregular)

    # Noise kernel
    ls_noise = pm.Gamma("ls_noise", alpha=2, beta=20.0)
    sigma_noise_w = pm.HalfNormal("sigma_noise_w", sigma=0.12)
    noise_kernel = 0.188**2 * pm.gp.cov.ExpQuad(input_dim=1, ls=ls_noise) + pm.gp.cov.WhiteNoise(sigma=sigma_noise_w)

    # Combine the kernels
    co2_kernel = long_term_trend_kernel + seasonal_kernel + irregularities_kernel + noise_kernel

with model:
    gp = pm.gp.Marginal(cov_func=co2_kernel)

    # sigma = pm.Deterministic("sigma", sigma_noise)
    y_ = gp.marginal_likelihood("y", X=X.reshape(-1, 1), y=y, sigma=sigma_noise)

    marginal_post = pm.sample(500, tune=200, target_accept=0.95, nuts_sampler="numpyro")

    # add the GP conditional to the model, given the new X values
    f_pred = gp.conditional("f_pred", X_test.reshape(-1, 1))

    # Predict y values at test points using the posterior samples
    y_pred = gp.conditional("y_pred", X_test.reshape(-1, 1), pred_noise=True)
    y_samples = pm.sample_posterior_predictive(marginal_post.sel(draw=slice(0, 20)), var_names=["y_pred"])

    pred_samples = pm.sample_posterior_predictive(marginal_post.sel(draw=slice(0, 20)), var_names=["f_pred"])

Alone, the PyMC implementation struggles to sample the posterior distribution of the function $f$ for the Mauna Loa CO2 dataset. This difficulty arises from the dataset's complexity and size. The PyMC implementation is more computationally expensive compared to the scikit-learn implementation, which proves more efficient for this type of dataset. The PyMC approach is better suited for smaller datasets, where the advantages of a full Bayesian framework can be fully realized.

Disadvantages of Gaussian Process Regression

• Building a model becomes computationally expensive with large amounts of training data. The implementation of the algorithm discussed here involves inverting the covariance matrix $\Sigma = K(X, X)$ using Cholesky factorization, which incurs a memory complexity of $O(n^2)$ and a computational complexity of $O(n^3)$. There has been work on GP models that include only a subset of the data (Gramacy and Apley, 2015), which can be implemented more efficiently (Gramacy et al., 2014).
• Finding kernels (similarity functions) that suit the data can be time-consuming, and it remains an active area of research.
• Another issue is that the model often assumes a stationary covariance function. However, for many problems, the characteristics of the covariance function need to change across different regimes. Gramacy and Lee (2008) developed a hybrid tree-Gaussian process model that allows the covariance function to adapt over the range of input data.